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[德] 奥斯曼 著 / 科学出版社 / 2009-01 / 精装
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国外数学名著系列(续1)(影印版)59:几何5(最小曲面)
《几何5:最小曲面(影印版59)》gathersinoneplateanoverviewofsomeofthemostexcitingdevelopments,presentedbyfiveoftheleadingcontributorstothosedevelopments.HirotakaFujimoto,whoobtainedthedefinitiveresultsontheGaussmapofminimalsurfaces,reportsonNevanlinnaTheory,andMinimalSurfaces,StefanHildebrandtprovidesanup-to-dateaccountofthePlateanproblemandrelatedboundary-valueproblems.DavidHoffmanaridHermannKarcherdescribethewealthofresultsonembeddedsurfacesfromthepastdecade,startingwithCostassurfaceandthesubsequentHoffman-Meeksexamples.Finally,LeonSimoncoversthePDEaspectofminimalsurfaces,withasurveyofknownresultsbothintheclassicalcaseofsurfacesandinthehigherdimensionalcase,Thebookwillbeveryusefulasareferenceandresearchguidetograduatestudentsandresearchersinmathematics.
Thetheoryofminimalsurfaceshasexpandedinmanydirectionsoverthepastdecadeortwo.
1.Introduction2.BasicTheoryandtheGlobalWeierstrassRepresentation2.1FiniteTotalCurvature2.2TheExampleofChen-Gackstatter2.3EmbeddednessandFiniteTotalCurvature:NecessaryConditions2.3.1Flux2.3.2Torque2.3.3TheHalfspaceTheorem2.4SummaryoftheNecessaryConditionsforExistenceofCompleteEmbeddedMinimalSurfaceswithFiniteTotalCurvature3.ExampleswithRestrictedTopology:ExistenceandRigidity3.1CompleteEmbeddedMinimalSurfacesofFiniteTotalCurvatureandGenusZero:theLopez-RosTheorem4.ConstructionoftheDeformationFamilywithThreeEnds4.1HiddenConformalSymmetries4.2TheBirdcageModel4.3MeromorphicFunctionsConstructedbyConformalMappings4.3.1TheFunctionTandItsRelationshiptou4.4TheFunctionzandtheEquationfortheRiemannSurfaceinTermsofzandu4.5TheWeierstrassData4.6TheLogarithmicGrowthRates4.7ThePeriodandEmbeddednessProblemsfortheSurfacesMk,x4.8TheDetailsoftheSolutionofthePeriodProblem,I.SimplificationoftheIntegrals4.9TheDetailsoftheSolutionofthePeriodProblem,II.TheMonotonicityLemma4.9.1ProofoftheMonotonicityLemma5.TheStructureoftheSpaceofExamples5.1TheSpaceofComplete,EmbeddedMinimalSurfacesofFiniteTotalCurvature5.2SomeQuestionsandConjectures6.FiniteTotalCurvatureVersusFiniteTopology6.1Complete,Properly-Immersed,MinimalSurfaceswithMoreThanOneEnd6.2CompleteEmbeddedMinimalSurfacesofFiniteTopologyandMoreThanOneEnd6.3CompleteEmbeddedMinimalSurfacesofFiniteTopologywithOneEnd7.StabilityandtheIndexoftheGaussMapReferences
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